MULTIPLIERLESS DESIGN OF BIORTHOGONAL DUAL-TREE COMPLEX WAVELETTRANSFORM USING LIFTING SCHEME

Adeel Abbas and Trac D. Tran

Department of Electrical and Computer Engineering,The Johns Hopkins University, Baltimore, MD 21218

E-mail: adeel@jhu. edu, trac@jhu.edu

ABSTRACT

In this work, we present the design, implementation and ap-plication of two families of biorthogonal dual-tree complexwavelet transform (CWT) filters using lifting scheme. Thefirst design is achieved using exhaustive search with codinggain, DC leakage and directional selectivity as the fundamen-tal criteria. The second set of filters are derived from thebiorthogonal design procedure that was recently suggestedby Selesnick. Furthermore, this paper also introduces a newtheorem that suggests that lifting implementation of filtersthat are time-reversals of each other is closely related. Vari-ous applications are presented to validate the proposed design

x

Fig. 1. Kingsbury's dual tree DWT (left). The Lifting schemeconstruction (right).

The lifting scheme [3] is a very popular architecture forbuilding fast and efficient signal decompositions. In matrixform the implementation for Fig. 1 (right), is written as

scheme, including pertormance in denoising as well as in the F1 K 0 1 0 1 P(z) 1 X1JPEG-2000 image coding standard. [ y2 ] [0 1/K [ U(z) ij [0 1 L2 ]

Index Terms- Wavelet transforms, transform coding. The inverse of lifting step is simple: we simply subtract outwhat was added in at the forward transform. Lifting scheme

1. INTRODUCTION is very robust to quantization and structurally enforces prefectreconstruction. Another advantage of lifting is that it can be

The dual-tree complex wavelet transform [1] has recently been used to build wavelet transforms that map integers to integers,a subject of considerable interest in the wavelet community. a very desirable feature for lossless image compression. ItShown in Fig. 1, the construction uses two separate max- has been shown in [3] that any FIR wavelet filter pair canimally decimated wavelet decompositions, each capable of be realized using lifting scheme. It is particularly attractiveperfect reconstruction. The {Ho (z), H1 (z)} filter decompo- when used in implementing symmetric/antisymmetric filters,sition provides real part of the transform, whereas the {Go (z), G1 (b4Qause in that case, all lifting polynomials can be simplifiedfilter pair generates the complex part of the transform. From to be of the form a (l ± z-1), where a is a scalar. LiftingSelesnick's work [2], it is now established that when these fil- scheme can be used in several other ways. For example it canters are designed such that they are offset by a half sample, realize lattice structures [3] and even M-channel filter banks.the resulting CWT wavelet bases form a Hilbert transformpair. In other words, for -w <w <w, if 2. DESIGN PROCEDURE

Go(w) - Ho(w) x e-j (w) 0(w) w/2,

Then~b(J)- i"~bh(WO) CO>O0

L iihh(W) C < 0

Where Vbh (w) and g (w) are the fourier transforms of waveletfunctions of the real and imaginary trees.

This work is partially supported by NSF CAREER CCF-0093262 andNSF CCF-0430869

The biorthogonal designs presented here are based on the spec-tral factorization approach described by Selesnick in [2]. Us-ing this procedure, filters with certain regularity and approxi-mation to the flat all pass delay filter can be constructed. Se-lesnick's construction enforces Go (z) to be a time reversal ofHo(z). Since there has to be a half sample delay betweenGo (z) and Ho (z), the filters cannot be linear phase filters.

We used the software Singularl to develop a framework to

'Algebraic Geometry Group, Univ. of Kaiserslautem, Germany -

1-4244-0481-9/06/$20.00 C2006 IEEE 1605 ICIP 2006

switch from filter domain to lifting domain and perform fullsearch in the lifting space. Smoother wavelets are needed forHo (z), because it will also be used in compression applica-tions. Therefore the quantization bits for Ho (z) are selectedso that wavelet smoothness is maximized. Since Go (z) is atime-reversal of Ho (z), as it will be explained shortly, its lift-ing implementation can be directly inferred from Ho (z) im-plementation. The overall design procedure is governed byoptimization of four parameters, which are:

* Coding Gain relates to the ability of a sub-band coder tocompress most of the signal energy in least number of bands.The biorthogonal coding gain Cg is defined as:

,T2C = lox log1o ( U 2 M

(n1

72 X llfi 112).,4~~~~H=.x\i

Wtnere:Ma2xCxi

11 fi 11 2

Number of sub-bandsVariance of inputVariance of i-th sub-bandL-2 norm of i-th synthesis basis function

Where input is a first order Gaussian-Markov process withzero-mean, unit variance and correlation coefficient 0.95.

* The directional selectivity of the dual-tree CWT pair isalso an important metric. Since the wavelet bases form a closeapproximation to the Hilbert transform pair, the frequency re-sponse of the function O9h [n] + j g [n] must have high atten-uation in the region -oc < w < 0. In order to measure this,we introduce Hilbert PSNR (HPSNR), defined as

HPSNR = 10 x log ( x h(W) + g(w) 2

* The DC leakage (DCPSNR) measures the amount ofenergy picked up in the high-pass sub-band when a constantsignal is presented as input. It is mathematically defined as

DCPSNR = 10 x log1o ( ZE h[nl

For the analysis high-pass filter h, [n], if >n hi [n] 0 thenthis implies that no DC will be picked up in the high-passsub-band. This property is directly related to the smoothnessof the resulting scaling and wavelet functions.

* The hardware complexity of an algorithm is generallydetermined by the number of additions and shifts required inits implementation. From a hardware perspective, adders areconsidered to be the most expensive modules in a CPU's arith-metic logic unit. Likewise, smaller amount of shifts allow thealgorithm to be mapped on a CPU with narrower bus width,a very attractive feature for portable devices where bus widthand battery power are limited.

http://www.singular.uni-k1.de/

2.1. Time Reversal Relationship In Lifting Domain

Consider Selesnick's biorthogonal designs, where the filterpair {Go(z), GI(z)} is related to {Ho(z), HI(z)} by a timereversal. If Uo(z), Ui (z), . . . UN 1(z) and Po (z), PI (z),* PM-I (z) are respectively the update and prediction poly-nomials that implement {Ho (z), H1 (z) }, then the polyno-mials that will implement {Go(z), Gi(z)} will be Uo(z-1),Ui(z 1), . ., UN-1(Z 1) andPo(z 1), PI(z 1),. PM 1(Z 1)Thus, by time-reversing all update/predict stages in a liftingscheme, we can implement time-reversed filters. Note thatto satisfy causality, some delay has to be introduced into sub-bands. This holds for odd-length as well as even-length filters,except that in even-length case, the polyphase matrix has tobe post-multiplied by anti-diagonal (reversal) matrix. Thus, ifHp(z) has lifting decomposition of the form

Hp(z) [0 IK ]lPi0(z)]jUi(z)

Then, GP (z) has to be of the form

GP(z) = A(z) [ II/K 1 17 Pi(z )Ui(z 1)

GP(z) has to be post-multiplied by [ 00 ] if filters are

even-length. Note that some delay has to be introduced tomatch data samples. The proof of the theorem is omitted herebecause of lack of space. 2

2.2. Design via Exhaustive Search

For a very general case, with one update and one predict stage,the polyphase matrices will be of the following form

H (z) [ - ° ] [ z+d0z-1 ] [ a+bz 1 ]

G() [- - 0 2 ] z-1 d+cz- 1) [ 1 0Gp(Z)-= o -1I/V'2 L ° z-l bL baz- 1 I -

By utilizing that >jn ho [n] = V2 and En hi [n] = 0, we canrepresent b and d in terms of a and c respectively,

a + b -1 (1)(2)c+d = 1/2.

The above conditions also enforce one zero at exactly z = 1(i.e. no DC component leaks into the high pass subband).Therefore, the filters can be designed using an exhaustive

2Interested readers will find this in "Multiplierless integer coeffcient dual-tree complex wavelet transform: design and implementation," IEEE Trans.on Signal Processing, Submitted.

1606

11 T1n

search that maximizes coding gain and Hilbert performanceagainst two free parameters, a and c. Fig. 2, shows the resultsfrom this search. Coding gain and Hilbert PSNR are plottedin 3D against a and c. The optimal area is marked with blackcylinder and it corresponds to values a =-3/4, b =-1/4,c = 1/8 and d = 3/8. This design is called Lift-5/3. It turnsout that Lift-5/3 is very close to the 5/3 filter pair that canbe designed using Selesnick's construction (called Sel-5/3).With one vanishing moment and all pass filter of order one,both of these are the simplest designs that can be achievedusing dual-tree CWT construction. Furthermore, if we startwith Selesnick's 5/3 filters and perform Euclidean factoriza-tion algorithm [3], the result is a solution that is very close tothe our 5/3 solution. As shown in Fig. 3, Lift-5/3 has one

GP(z) [ 322[() 01 o1Q7+ -2)1

z-' ' (7 + 2z-2)

-1 Z1(4 -3)]O z-1 ...

[ -'(I+3z-1)ILift-9/3 filter pair was developed by approximating coeffi-cients from 9/3 biorthogonal design using Selesnick's scheme(The original filters are termed Sel-9/3).

Table 1. Quantized filter coefficients for proposed designsLift-5/3 Lift-9/3

n ho [n] hi [n] ho [n] hi [n]1 0 0 -3/2 02 3/25 0 1/26 03 - 1/23 0 23/28 04 -11/24 -3/22 -1/23 05 -3/23 1 -41/26 -3/226 3/25 1X22 -7/24 17 0 0 1/24 -1/228 0 0 1/24 09 _ _ -1/26 010 U UK 2 -1/ 2X -1/

3. EVALUATIONS AND APPLICATIONS

Fig. 2. Exhaustive search results for Coding Gain (left) andHilbert PSNR (right). The black cylinders mark the point thatcorresponds to Lift-5/3 implementation.

update and one prediction stage, followed by a scaling factor.Like the Daubechies 5/3 filter set, Lift-5/3 is very efficient. Asexpected, the lifting polynomials for both the decompositionsare related.

2 X62

l442132

44/w321/ r~ 1,83+lIt

I ~~~~~~-2332D

Fig. 3. Lift-5/3, {Ho, H1} (left) and {Go, G1} (right).

2.3. Design via Approximation

The second approximation is called Lift-9/3. Lift-9/3 uses thesame high-pass filter as the 5/3, but improves on the numberof vanishing moments of the low-pass filter. Table 1 showsfilter coefficients for this approximation. The polyphase fac-torization will be given by

Hp(z)

x- z

01'

44o

1[4(1±4z-3)]

+ -2) l -- I

z-l 'L-(3+ z-') I2

In this section, we will review some of the applications thatdemonstrate performance of the proposed filters.

3.1. Performance on JPEG-2000

As stated, the two factors that directly affect the compres-sion performance of a wavelet system are coding gain and DCleakage. In Table 3 (A) and (C), we compare coding gain andDC leakage of our implementations in terms of PSNR. Theseresults indicate that our filters should exhibit satisfactory per-formance in both lossy as well as lossless compression ap-plications. Fig. 4 plots the wavelet and scaling functions ofLift-5/3.

The proposed families of wavelet filters have been imple-mented in JPEG-2000 compression engine. Table 2 comparesPSNR results of our filters with popular Daubechies filtersand the unquantized designs. In summary, the performanceof quantized filters is comparable to that of the Daubechies'filters and better than previously published dual-tree CWT fil-ters with irrational-coefficients.

2- 0=1

O 5 O 5 ~~~~O5_a50

(a) (b) (c) (d)6 [

Fig. 4. Lift-5/3: (a) Oh [n] (b) Oh [n] (C) of [n] (d) of [n]

1607

I

Table 2. JPEG-2000 performance on standard test imagesBitrate Daub-9/7 Daub-5/3 Sel-5/3 Lift-5/3

Lena 1 40.34 39.52 37.19 38.970.5 37.22 36.40 34.68 35.610.25 34.04 33.26 32.07 32.510.125 30.65 30.11 29.24 29.52

Barbara 1 37.18 35.87 33.01 35.080.5 32.30 30.82 29.39 30.230.25 28.41 27.45 26.49 27.000.125 25.26 24.73 24.25 24.50

Boat 1 39.30 38.72 35.11 38.200.5 34.64 34.06 32.34 33.700.25 31.06 30.59 29.56 30.190.125 28.01 27.97 27.29 27.61

3.2. Hilbert Performance and Directional 2-D Wavelets

A consequence of good Hilbert transform properties is bet-ter directional selectivity in the 2D domain. The separable2D wavelets can only capture horizontal, vertical or diagonaledges (00, 450 and 900). The dual-tree, on the other handcan find edges at several other angles (15°, 750, 1050, 135°,1650). These edges are obtained by adding and subtractingwavelet coefficients from real and imaginary trees. Fig. 5plots the impulse response of wavelets for the Lift-5/3 de-sign. As stated earlier, the IHPSNR directly measures this di-rectional selectivity (results are plotted in Table 3).

Fig. 5. Directional 2D Wavelets for Lift-513. From left toright, edges at: 150, 750, 450, 1650, 1050, 1350.

Fig. 6. The noisy image was denoised using Lift-9/3.

3.3. Denoising performance

Fig. 7 shows3 PSNR versus Threshold point plot for the stone-henge image. The integer-coefficient designs perform as goodas the originals. Fig. 6 illustrates another example of denois-ing performed using Lift-9/3 filters.

3Most of our denoising experiments were based on the Matlab code avail-able at http://taco.poly.edu/WaveletSoftware/index.htmI

32.77

32.765

zL3)n. 32.76-

Lift-5/3Lift-9/332.755

1.3 1.4 1.5 1.6 1.7 1.8 1.9 2Threshold points

Fig. 7. PSNR versus Threshold points plot.

Table 3. Property comparisons of the proposed designs

A. Coding Gain B. HPSNR C. DCPSNR(dB) (dB) (dB)

Filters Ho Fo Go Po Ho GoSel-5/3 9.29 6.95 9.29 6.95 38.83 207.6 208.9Lif-5/3 9.17 7.69 9.17 7.69 33.20 00 00Sel-9/3 9.20 8.78 9.20 8.78 41.31 207.6 208.9Lift-9/3 9.22 8.52 9.22 8.52 37.58 00 00

4. CONCLUSION

This paper presents two multiplierless, lifting scheme imple-mentations of the dual-tree CWT. Since the lifting polynomi-als were chosen to be dyadic rationals, a very efficient im-plementation with only shifts and additions is realizable. Anew theorem was introduced, which claims that filters thatare time-reversals of each other are also related in the liftingdomain. The new designs have been evaluated with variousperformance measures, including the JPEG-2000 lossy imagecompression standard and image denoising.

5. REFERENCES

[1] N. G. Kingsbury, "The dual-tree complex wavelet trans-form: a new technique for shift invariance and directionalfilters," IEEE DSP Workshop 98, Aug. 1998.

[2] I. W. Selesnick, "The design of approximate Hilberttransform pairs of wavelet bases," IEEE Trans. on Sig-nal Processing, vol. 50,, pp. 1144-1152, May 2002.

[3] I. Daubechies and W. Sweldens, "Factoring wavelettransforms in lifting step," Bell Labs., Lucent Technolo-gies, Murray Hill, NJ, Tech. Rep., 1996.

[4] A. Abbas and T. D. Tran, "Fast approximations of theorthogonal dual-tree wavelet bases," Proc. IEEE Interna-tional Conference on Acoustics, Speech and Signal Pro-cessing, pp. 605-608, March 2005.

1608